A necessary and sufficient condition for all the eigenvalues of a real 
matrix 
to have negative real
parts is that the equation
 |
has as a solution where 
is an 
matrix and 
is a positive
definite quadratic form.
Lyapunov's First Theorem
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1122, 2000.Referenced on Wolfram|Alpha
Lyapunov's First TheoremCite this as:
Weisstein, Eric W. "Lyapunov's First Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LyapunovsFirstTheorem.html